Use labels to reference line numbers of algorithm.

[hpcc2014.git] / hpcc.tex

diff --git a/hpcc.tex b/hpcc.tex
index dd06eebc905d58734f2d3d5c16022c85850180be..3dd67ef8614cc072037465711a8576e58d6263e2 100644 (file)
--- a/hpcc.tex
+++ b/hpcc.tex
@@ -108,23 +108,23 @@ network, etc.) but also a non-negligible CPU execution time. We consider in this
 parallel algorithms called \emph{numerical iterative algorithms} executed in a distributed environment. As their name
 suggests, these algorithms solve a given problem by successive iterations ($X_{n +1} = f(X_{n})$) from an initial value
 $X_{0}$ to find an approximate value $X^*$ of the solution with a very low residual error. Several well-known methods
 parallel algorithms called \emph{numerical iterative algorithms} executed in a distributed environment. As their name
 suggests, these algorithms solve a given problem by successive iterations ($X_{n +1} = f(X_{n})$) from an initial value
 $X_{0}$ to find an approximate value $X^*$ of the solution with a very low residual error. Several well-known methods

-demonstrate the convergence of these algorithms \cite{BT89,Bahi07}. 
+demonstrate the convergence of these algorithms~\cite{BT89,Bahi07}.

 
 Parallelization of such algorithms generally involve the division of the problem into several \emph{blocks} that will
 be solved in parallel on multiple processing units. The latter will communicate each intermediate results before a new
 iteration starts and until the approximate solution is reached. These parallel  computations can be performed either in
 \emph{synchronous} mode where a new iteration begins only when all nodes communications are completed,
 or in \emph{asynchronous} mode where processors can continue independently with few or no synchronization points. For
 
 Parallelization of such algorithms generally involve the division of the problem into several \emph{blocks} that will
 be solved in parallel on multiple processing units. The latter will communicate each intermediate results before a new
 iteration starts and until the approximate solution is reached. These parallel  computations can be performed either in
 \emph{synchronous} mode where a new iteration begins only when all nodes communications are completed,
 or in \emph{asynchronous} mode where processors can continue independently with few or no synchronization points. For

-instance in the \textit{Asynchronous Iterations~-- Asynchronous Communications (AIAC)} model \cite{bcvc06:ij}, local
+instance in the \textit{Asynchronous Iterations~-- Asynchronous Communications (AIAC)} model~\cite{bcvc06:ij}, local

 computations do not need to wait for required data. Processors can then perform their iterations with the data present
 at that time. Even if the number of iterations required before the convergence is generally greater than for the
 synchronous case, AIAC algorithms can significantly reduce overall execution times by suppressing idle times due to
 computations do not need to wait for required data. Processors can then perform their iterations with the data present
 at that time. Even if the number of iterations required before the convergence is generally greater than for the
 synchronous case, AIAC algorithms can significantly reduce overall execution times by suppressing idle times due to

-synchronizations especially in a grid computing context (see \cite{Bahi07} for more details).
+synchronizations especially in a grid computing context (see~\cite{Bahi07} for more details).

 
 Parallel numerical applications (synchronous or asynchronous) may have different configuration and deployment
 requirements.  Quantifying their resource allocation policies and application scheduling algorithms in
 grid computing environments under varying load, CPU power and network speeds is very costly, very labor intensive and very time
 
 Parallel numerical applications (synchronous or asynchronous) may have different configuration and deployment
 requirements.  Quantifying their resource allocation policies and application scheduling algorithms in
 grid computing environments under varying load, CPU power and network speeds is very costly, very labor intensive and very time

-consuming \cite{Calheiros:2011:CTM:1951445.1951450}. The case of AIAC algorithms is even more problematic since they are very sensible to the
+consuming~\cite{Calheiros:2011:CTM:1951445.1951450}. The case of AIAC algorithms is even more problematic since they are very sensible to the

 execution environment context. For instance, variations in the network bandwidth (intra and inter-clusters), in the
 number and the power of nodes, in the number of clusters... can lead to very different number of iterations and so to
 very different execution times. Then, it appears that the use of simulation tools to explore various platform
 execution environment context. For instance, variations in the network bandwidth (intra and inter-clusters), in the
 number and the power of nodes, in the number of clusters... can lead to very different number of iterations and so to
 very different execution times. Then, it appears that the use of simulation tools to explore various platform


@@ -138,7 +138,7 @@ best of execution time.
 
 To our knowledge, there is no existing work on the large-scale simulation of a real AIAC application. The aim of this
 paper is twofold. First we give a first approach of the simulation of AIAC algorithms using a simulation tool (i.e. the
 
 To our knowledge, there is no existing work on the large-scale simulation of a real AIAC application. The aim of this
 paper is twofold. First we give a first approach of the simulation of AIAC algorithms using a simulation tool (i.e. the

-SimGrid toolkit \cite{SimGrid}). Second, we confirm the effectiveness of asynchronous mode algorithms by comparing their
+SimGrid toolkit~\cite{SimGrid}). Second, we confirm the effectiveness of asynchronous mode algorithms by comparing their

 performance with the synchronous mode. More precisely, we had implemented a program for solving large non-symmetric
 linear system of equations by numerical method GMRES (Generalized Minimal Residual) []. We show, that with minor
 modifications of the initial MPI code, the SimGrid toolkit allows us to perform a test campaign of a real AIAC
 performance with the synchronous mode. More precisely, we had implemented a program for solving large non-symmetric
 linear system of equations by numerical method GMRES (Generalized Minimal Residual) []. We show, that with minor
 modifications of the initial MPI code, the SimGrid toolkit allows us to perform a test campaign of a real AIAC


@@ -160,7 +160,7 @@ carried out will be presented before some concluding remarks and future works.
 
 As exposed in the introduction, parallel iterative methods are now widely used in many scientific domains. They can be
 classified in three main classes depending on how iterations and communications are managed (for more details readers
 
 As exposed in the introduction, parallel iterative methods are now widely used in many scientific domains. They can be
 classified in three main classes depending on how iterations and communications are managed (for more details readers

-can refer to \cite{bcvc06:ij}). In the \textit{Synchronous Iterations~-- Synchronous Communications (SISC)} model data
+can refer to~\cite{bcvc06:ij}). In the \textit{Synchronous Iterations~-- Synchronous Communications (SISC)} model data

 are exchanged at the end of each iteration. All the processors must begin the same iteration at the same time and
 important idle times on processors are generated. The \textit{Synchronous Iterations~-- Asynchronous Communications
 (SIAC)} model can be compared to the previous one except that data required on another processor are sent asynchronously
 are exchanged at the end of each iteration. All the processors must begin the same iteration at the same time and
 important idle times on processors are generated. The \textit{Synchronous Iterations~-- Asynchronous Communications
 (SIAC)} model can be compared to the previous one except that data required on another processor are sent asynchronously


@@ -169,10 +169,10 @@ but unfortunately, the overlapping is only partial and important idle times rema
 computing context, where the number of computational nodes is large, heterogeneous and widely distributed, the idle
 times generated by synchronizations are very penalizing. One way to overcome this problem is to use the
 \textit{Asynchronous Iterations~-- Asynchronous Communications (AIAC)} model. Here, local computations do not need to
 computing context, where the number of computational nodes is large, heterogeneous and widely distributed, the idle
 times generated by synchronizations are very penalizing. One way to overcome this problem is to use the
 \textit{Asynchronous Iterations~-- Asynchronous Communications (AIAC)} model. Here, local computations do not need to

-wait for required data. Processors can then perform their iterations with the data present at that time. Figure
-\ref{fig:aiac} illustrates this model where the gray blocks represent the computation phases, the white spaces the idle
+wait for required data. Processors can then perform their iterations with the data present at that time. Figure~\ref{fig:aiac}
+illustrates this model where the gray blocks represent the computation phases, the white spaces the idle

 times and the arrows the communications. With this algorithmic model, the number of iterations required before the
 times and the arrows the communications. With this algorithmic model, the number of iterations required before the

-convergence is generally greater than for the two former classes. But, and as detailed in \cite{bcvc06:ij}, AIAC
+convergence is generally greater than for the two former classes. But, and as detailed in~\cite{bcvc06:ij}, AIAC

 algorithms can significantly reduce overall execution times by suppressing idle times due to synchronizations especially
 in a grid computing context.
 
 algorithms can significantly reduce overall execution times by suppressing idle times due to synchronizations especially
 in a grid computing context.
 


@@ -277,8 +277,8 @@ is solved independently by a cluster and communications are required to update t
 \For {$k=0,1,2,\ldots$ until the global convergence}
 \State Restart outer iteration with $x^0=x^k$
 \State Inner iteration: \Call{InnerSolver}{$x^0$, $k+1$}
 \For {$k=0,1,2,\ldots$ until the global convergence}
 \State Restart outer iteration with $x^0=x^k$
 \State Inner iteration: \Call{InnerSolver}{$x^0$, $k+1$}

-\State Send shared elements of $X_l^{k+1}$ to neighboring clusters
-\State Receive shared elements in $\{X_m^{k+1}\}_{m\neq l}$
+\State\label{algo:01:send} Send shared elements of $X_l^{k+1}$ to neighboring clusters
+\State\label{algo:01:recv} Receive shared elements in $\{X_m^{k+1}\}_{m\neq l}$

 \EndFor
 
 \Statex
 \EndFor
 
 \Statex


@@ -303,9 +303,9 @@ $\{A_{lm}\}_{m\neq l}$ are off-diagonal matrices of sparse matrix $A$ and
 $\{X_m\}_{m\neq l}$ contain vector elements of solution $x$ shared with
 neighboring clusters. At every outer iteration $k$, asynchronous communications
 are performed between processors of the local cluster and those of distant
 $\{X_m\}_{m\neq l}$ contain vector elements of solution $x$ shared with
 neighboring clusters. At every outer iteration $k$, asynchronous communications
 are performed between processors of the local cluster and those of distant

-clusters (lines $6$ and $7$ in Figure~\ref{algo:01}). The shared vector
-elements of the solution $x$ are exchanged by message passing using MPI
-non-blocking communication routines.
+clusters (lines~\ref{algo:01:send} and~\ref{algo:01:recv} in
+Figure~\ref{algo:01}). The shared vector elements of the solution $x$ are
+exchanged by message passing using MPI non-blocking communication routines.

 
 \begin{figure}[!t]
 \centering
 
 \begin{figure}[!t]
 \centering